Proper velocity and frame-invariant acceleration in special relativity

Abstract

We examine here a possible endpoint of the
trends, in the teaching literature, away from use of relativistic masses
(such as m' = gamma m in the momentum = mass
times velocity expression) and toward use of proper velocity
dx/dto = gamma v (e.g. in that same expression).
We show that proper time & proper velocity, taken as components of a
non-coordinate time/velocity pair, allow one to introduce time dilation and
frame-invariant acceleration/force 3-vectors in the context of one
inertial frame, before subjects involving multiple frames (like Lorentz
transforms, length contraction, and frame-dependent simultaneity) need be
considered. We further show that many post-transform equations (like the
velocity-addition rule) acquire elegance and/or utility not found in the
absence of these variables.

Introduction

Efforts to connect classical and relativistic concepts will be with us as
long as classical kinematics is taught to introductory students. For
example, the observation that relativistic objects behave at high speed as
though their inertial mass increases in the p=mv expression, led to the definition (used in many early
textbooks {e.g. French}) of relativistic mass m' = mgamma.
Such efforts might help to: (A) get the most from first-taught
relationships, and (B) find what is fundamentally true in both classical
and relativistic approaches. The concepts of transverse (m') and longitudinal (m'' = mgamma^3) masses have
similarly been used {e.g. Blatt} to preserve relations of the form Fx = max
for forces perpendicular and parallel, respectively, to the velocity
direction.

Unfortunately for these relativistic masses, no deep sense in which mass
either changes or has directional dependence has emerged. They put familiar
relationships to use in keeping track of non-classical behaviors (item A
above), but do not (item B above) provide frame-invariant insights or make
other relationships simpler as well. As a result, majority acceptance of
their use seems further away now than it did several decades ago
{Goldstein}.

A more subtle trend in the pedagogical literature has been toward the
definition of a quantity called proper velocity {cf. SearsBrehme,Shurcliff},
which can be written as w = gammav.
We use the symbol w here because it is not in common use elsewhere in
relativity texts, and because w looks like gammav from a distance.
This quantity also lets us cast the momentum expression above in classical
form as a mass times a velocity, ie. as p = mw.
Hence it serves at least one of the ``type A'' goals served by m' above.

We show here that, when introduced as part of a non-coordinate time/velocity
pair in pre-transform special relativity, proper velocity allows us to
introduce relativistic momentum, time-dilation, and frame-invariant
relativistic acceleration in context of a single inertial frame. Moreover,
through use of proper velocity many relationships (including post-transform
relationships like velocity addition which require consideration of multiple
frames) are made simpler and sometimes more useful. Hence it appears to
serve goals of ``type B'' mentioned above as well.

One object, one frame, but two times: the consequences

One may argue that a fundamental break between classical and relativistic
kinematics involves the observation that time passes differently for moving
observers, than for stationary ones. To quantify this, we define two time
variables when describing the motion of a single object (traveler) with
respect to a single inertial coordinate frame. These are the coordinate time
t and the proper (or traveler) time to. Note that the proper time may
be different for different travelers.

It follows from above that we might also consider two velocities, namely
coordinate velocity v = dx/dt, and
proper (or traveler-kinematic) velocity w = dx/dto.
It is helpful to distinguish the units used to
measure these velocities, by saying that the first measures distance
traveled per unit coordinate time, while the latter measures distance
traveled per unit traveler time. Convenient units are [lightyears per
coordinate year] and [lightyears per traveler year], respectively. Each of
these velocities can be calculated from the other by knowing the
velocity-dependence of the ``traveler's speed of time'' gamma=dt/dto, via
the equation

(1)

Note that all displacements dx are defined with respect to a single
inertial (unprimed) frame. Thus proper velocity is not simply a coordinate
velocity measured with respect to a different frame. It is rather one of an
infinite number of non-coordinate velocities, definable in context of
time/velocity pairs experienced by one of the infinite number of observers
who may choose to track the motion of our given object on a given map using
their own clock{Noncoord}. The cardinal rule here is:
measure all displacements
from the vantage point of the chosen inertial (or ``map'') frame. Thus
proper velocity w is simply the rate at which the
coordinates of the traveler change per unit traveler time on a map of
the universe (e.g. in the traveler's ``glove compartment'') which was drawn
from the point of view of the chosen inertial reference frame, and not from
the point of view of whatever inertial frame the traveler happens to be in
at a given time.

A number of useful relationships for this ``traveler's speed of time'',
including the familiar relationship to coordinate velocity, follow simply
from the (3+1)D nature of the flat spacetime metric (specifically from the
frame-invariant dot-product rule applied to the displacement and velocity
4-vectors). Their derivation therefrom is outlined in the
Appendix. For
introductory students, we can simply quote Einstein's prediction that
spacetime is tied together so that instead of gamma =1, one has gamma
=1/Sqrt[1-(v/c)^2}=E/mc^2, where E is Einstein's ``relativistic energy''
and c is the speed of light. It then follows from above that:

(2)

Here of course K is the kinetic energy of motion, equal classically to (1/2)mv^2.
One of the simplest exercises a student might do at this
point is to show that since v = w/gamma =
w/Sqrt[1+(w/c)^2], as proper
speed w goes to infinity, the coordinate speed v never gets larger
than c! All of classical kinematics also follows from these things
if and only if gamma is near 1, which as one can see above is true only
for speeds v much less than c.

Given these tools to describe the motion of an object with respect to single
inertial reference frame, perhaps the easiest type of relativistic problem
to solve is that of time dilation. From the very definition of gamma as a
``traveler's speed of time'', and the velocity relations which show that
gamma is greater than or equal to 1, it is easy to see that the traveler's clock will always run
slower than coordinate time. Hence if the traveler is going at a constant
speed, one has from equation (2) that traveler time is dilated
(spread out over a larger interval) relative to coordinate time, by the
relation

(3)

In this way time-dilation problems can be addressed, without first
introducing the deeper complications (like frame-dependent simultaneity)
associated with multiple inertial frames. This is the first of several
skills that this strategy can offer to students taking only introductory
physics, a ``type A'' benefit according to the introduction. A practical
awareness of the non-global nature of time thus need not require a
readiness for the abstraction of Lorentz transforms.

Equations (2) above also allow one to relate these velocities to
energy. Hence an important part of relativistic dynamics is in hand as well.
Another important part of relativistic dynamics, as mentioned in the
introduction, takes on a familiar form since momentum at any speed is

(4)

This relation has important scientific consequences as well. It shows that
momentum like proper velocity has no upper limit, and that coordinate
velocity becomes irrelevant to tracking momentum at high speeds.

One might already imagine that proper speed w is the important speed to a
relativistic traveler trying to get somewhere with minimum traveler time.
Equation (4) shows that it is also a more interesting speed from
the point of view of law enforcement officials wishing to minimize
fatalities on a highway in which relativistic speeds are an option. This a
``type B'' benefit of proper velocity. In this context, it is not surprising
that the press doesn't report ``land speed record'' for the fastest
accelerated particle. New progress only changes the value of v in the 7th
or 8th decimal place. The story of increasing proper velocity, in the
meantime, goes untold to a public whose imagination might be captured
thereby. After all, the educated lay public (comprised of those who have had
only one physics course) is by and large under the impression that the
lightspeed limit rules out major progress along these lines.

The frame-invariant acceleration 3-vector

The foregoing relations introduce, in context of a single inertial frame and
without Lorentz transforms, many of the kinematical and dynamical relations
of special relativity taught in introductory courses, in modern physics
courses, and perhaps even in some relativity courses. In this section, we
cover less familiar territory, namely the equations of relativistic
acceleration. Forces if defined simply as rates of momentum change in
special relativity have no frame-invariant formulation, and hence Newton's
2nd Law retains it's elegance only if written in coordinate-independent
4-vector form. It is less commonly taught, however, that a
frame-invariant 3-vector acceleration can be defined (again also in context
of a single inertial frame). We show that, in terms of proper velocity and
proper time, this acceleration has three simple integrals when held
constant. Moreover, it bears a familiar relationship to the special
frame-independent rate of momentum change felt by an accelerated
traveler.

By again examining the frame-invariant scalar product of a 4-vector (this
time of the acceleration 4-vector), one can show (as we do in the Appendix)
that a ``proper acceleration'' for a given object, which is the same to all
inertial observers and thus ``frame-invariant'', can be written in terms of
components for the classical acceleration vector (i.e. the second
coordinate-time t derivative of displacement x) by:

(5)

This is quite remarkable, given that a is so
strongly frame-dependent! Here the ``transverse time-speed" gammaperp is defined as 1/Sqrt[1-(vperp/c)^2], where vperp is the
component of coordinate velocity v perpendicular to the
direction of coordinate accceleration a. In this section
generally, in fact, subscripts || and perp refer to parallel
and perpendicular component directions relative to the direction of coordinate
acceleration a, and not
(for example) relative to coordinate velocity v.

Before considering integrals of the motion for constant proper acceleration
alpha, let's review the classical integrals of motion
for constant acceleration a. These can be written as
a = Delta v||/Delta t = (1/2) Delta(v^2)/Delta x||. The
first of these is associated with conservation of momentum in the
absence of acceleration, and the second with the work-energy theorem. These
may look more familiar in the form v||f = v||i + a Delta
t, and v||f^2 = v||i^2 + 2 a Delta x||.
Given that coordinate velocity has an upper limit at the speed of light, it
is easy to imagine why holding coordinate acceleration constant in
relativistic situations requires forces which change even from the
traveler's point of view, and is not possible at all for Delta
t greater than (c-v||i)/a.

Provided that proper time to, proper velocity w, and
time-speed gamma
can be used as variables, three simple integrals of the proper
acceleration can be obtained by a procedure which works for integrating
other non-coordinate velocity/time expressions as well{Noncoord}. The
resulting integrals are summarized in compact form, like those above, as

(6)

Here the integral with respect to proper time to has been simplified by
further defining the hyperbolic velocity angle or rapidity{TaylorWheeler}
eta|| = ArcSinh[w||/c] = ArcTanh[v||/c]. Note that both vperp and the
``transverse time-speed'' gammaperp are constants, and hence both
proper velocity, and longitudinal momentum p|| = mw||, change at a uniform rate when
proper acceleration is held constant. If motion is only in the direction of acceleration, gammaperp is 1, and Delta p/Delta t = m alpha in the classical tradition.

In order to visualize the relationships defined by equation 6,
it is helpful to plot for the (1+1)D or gammaperp=1 case all
velocities and times versus x in dimensionless form from a common origin
on a single graph (i.e. as v/c, alphato/c,
w/c=alpha t/c, and
gamma versus alpha x/c^2). As shown in Fig. 1, v/c is asymptotic to
1, alpha to/c is exponential for large arguments,
w/c=alpha t/c are
hyperbolic, and also tangent to a linear gamma for large arguments. The
equations underlying this plot, from eqn 6 for gammaperp=1
and coordinates sharing a common origin, can be written simply as:

(7)

This universal acceleration plot, adapted to the relevant range of
variables, can be used to illustrate the solution of, and possibly to
graphically solve, any constant acceleration problem. Similar plots
can be constructed for more complicated trips (e.g. accelerated twin-paradox
adventures) and for the (3+1)D case as well{Noncoord}.

In classical kinematics, the rate at which traveler energy E increases
with time is not frame-independent, but the rate at which momentum
p increases is invariant. In special relativity, these rates (when figured with
respect to proper time) relate to each other as time and space components,
respectively, of the acceleration 4-vector. Both are frame-dependent
at high speed. However, we can define proper force
separately as the force felt by an accelerated object. We show in the
Appendix that this is simply Fo = malpha. That is, all accelerated objects feel
a frame-invariant 3-vector force Fo in the direction of their acceleration.
The magnitude of this force can be calculated from any inertial frame, by multiplying the rate
of momentum change in the acceleration direction times gammaperp,
or by multiplying mass times the proper acceleration alpha. The classical
relation F = dp/dt = mdv/dt = md^2x/dt^2 = ma then becomes:

(21)

Even though the rate of momentum change joins the rate of energy change in
becoming frame-dependent at high speed, Newton's 2nd Law for 3-vectors thus retains a frame-invariant form.

Although they depend on the observer's inertial frame, it is instructive
to write out the rates of momentum and energy change in terms of the proper
force magnitude Fo. The classical equation relating rates of momentum
change to force is dp/dt = F = mai||, where i|| is
the unit vector in the direction of acceleration. This becomes

(8)

Note that if there are non-zero components of velocity in directions
both parallel and perpendicular to the direction of acceleration, then
momentum changes are seen to have a component perpendicular to the
acceleration direction, as well as parallel to it. These transverse momentum
changes result because transverse proper velocity
wperp = gamma vperp
(and hence momentum pperp) changes when traveler gamma
changes, even though vperp is staying constant.

As mentioned above, the rate at which traveler energy increases with time
classically depends on traveler velocity through the relation
dE/dt = Fv|| = ma*v.
Relativistically, this becomes

(9)

Hence the rate of traveler energy increase is in form very similar to that
in the classical case.

Similarly, the classical relationship between work, force, and impulse can
be summarized with the relation
dE/dx|| = F = dp||/dt.
Relativistically, this becomes

(10)

Once again, save for some changes in scaling associated with the
``transverse time-speed'' constant gammaperp, the form of the
classical relationship between work, force, and impulse is preserved in the
relativistic case. Since these simple connections are a result, and not the
reason, for our introduction of proper time/velocity in context of a single
inertial frame, we suspect that they provide insight into relations that are
true both classically and relativistically, and thus are benefits of ``type
B'' discussed in the introduction.

Proper-velocity equations in ``post-transform'' relativity

The foregoing treats calculations made possible, and analogies with
classical forms which result, if one introduces the proper time/velocity
variables in context of a single inertial frame, prior to discussion of
multiple inertial frames and hence prior to the introduction of Lorentz
transforms. Are the Lorentz transform, and other post-transform relations,
similarly simplified or extended? The answer is yes, although our insights
in this area are limited by the facts that: (i) we have taken only a cursory
look at post-transform material, and (ii) one key expansion area, the
treatment of acceleration, is already taken care of by the pre-transform
material above.

The Lorentz transform itself is simplified, in that it can be written using
proper velocity in the symmetric matrix form:

(11)

This seems to us an improvement over the asymmetric equations normally used,
but of course requires a bit of matrix and 4-vector notation that your
students may not be ready to exploit.

The expression for length contraction, namely L = Lo/gamma, is not
changed at all. The developments above do suggest that the concept of proper
length Lo, as the length of a yardstick in the frame in which it is at
rest, may have broader use as well. The relativistic Doppler effect
expression, given as frequency f = fo Sqrt[{1+(v/c)}/{1-(v/c)}] in terms of
coordinate velocity, also simplifies to f = fo/{gamma-(w/c)}.

The most noticeable effect of proper velocity, in the post-transform
relativity we've considered so far, involves simplification and
symmetrization of the velocity addition rule. The rule for adding coordinate
velocities v' and v to get
relative coordinate velocity v'' in inherently complicated, namely
v||'' =
(v'+v||)/(1+v||v'/c^2)
with vperp'' NOT equal to vperp. Here
subscripts refer to component orientation with respect to the direction
of v'. Moreover, for
high speed calculations, the answer is usually un-interesting since large
coordinate velocities always add up to something very near to c. By
comparison, if one adds proper velocities w' = gamma' v'
and w = gammav to get relative proper velocity w'',
one finds simply that the coordinate velocity factors
add while the gamma-factors multiply, i.e.

(12)

Note that the components transverse to the direction of v' are unchanged.

Physically more interesting questions can be answered with equation (12)
than with the coordinate velocity addition rule. For example, one
might ask what relative proper velocity (and hence momentum) is attainable
with colliding beams from an accelerator able to produce particles of proper
speed w for impact onto a stationary target. If one is using 50GeV
electrons in the LEP2 accelerator at CERN, gamma and gamma'
are E/mc^2 = 50GeV/511keV or 10^5, v and v' are essentially
c, and w and w' are hence 10^5c. Upon collision, equation
(12) tells us that in a collider the relative proper speed
w'' is (10^5)^2(c+c) = 2 x 10^10 c. Investment in a
collider thus buys a factor of 2 gamma = 2 x 10^5 increase in the
momentum (and energy) of collision. Compared to the cost of building a
10PeV accelerator for the equivalent effect on a stationary target, the
collider is a bargain indeed!

Conclusions

We show in this paper that introduction of two variables in the context of a
single inertial frame, specifically the non-coordinate proper (or
traveler-kinematic) time/velocity pair, lets students tackle time dilation
and relativistic acceleration problems, prior to consideration of issues
involving multiple inertial frames (like Lorentz transforms, length
contraction, and frame-dependent simultaneity). The cardinal rule to follow
when doing this is simple: All distances must be defined with respect to
``maps'' drawn from the vantage point of a single inertial reference frame.

We show further that a frame-invariant proper acceleration 3-vector
has three simple integrals of the motion, in terms of these variables. Hence
students can speak of the proper acceleration and force 3-vectors for an
object in frame-independent terms, and solve relativistic constant
acceleration problems much as they now do for non-relativistic problems in
introductory courses.

We further show that the use of these variables does more than
``superficially preserve classical forms''. Not only are more than one
classical equation preserved with minor change with these variables. In
addition, more interesting physics is accessible to students more quickly
with the equations that result. The relativistic addition rule for proper
velocities is a special case of the latter in point. Hence we argue that the
trend in the pedagogical literature, away from relativistic masses and
toward use of proper time and velocity in combination, may be a robust one
which provides: (B) deeper insight, as well as (A) more value from lessons
first-taught.

Acknowledgments

This work has benefited indirectly from support by the U.S. Department of
Energy, the Missouri Research Board, as well as Monsanto and MEMC Electronic
Materials Companies. It has benefited most, however, from the interest and
support of students at UM-St. Louis.

{TaylorWheeler} F. Taylor and J. A. Wheeler, Spacetime Physics
(W. H. Freeman, San Francisco, 1963 and 1992).
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